Краткое описание

The semicircle as base element. And the largest inscribed equilateral triangle.

General case

Segments in the general case

0) The radius of the semicircle: ${\displaystyle r_{0}}$
1) The side length of the inscribed equilateral triangle: ${\displaystyle a_{1}={\frac {2}{\sqrt {3}}}r_{0}\quad }$, see Calculation 1

Perimeters in the general case

0) Perimeter of base semicircle: ${\displaystyle P_{0}=2\cdot r_{0}+\pi \cdot r_{0}=(2+\pi )\cdot r_{0}\quad \approx 5.142\cdot r_{0}}$
1) Perimeter of inscribed equilateral triangle: ${\displaystyle P_{1}=3\cdot a_{1}=3\cdot {\frac {2}{\sqrt {3}}}r_{0}=2{\sqrt {3}}\cdot r_{0}\quad \approx 3.464\cdot r_{0}}$
S) Sum of perimeters: ${\displaystyle P_{s}=(2+\pi +2{\sqrt {3}})\cdot r_{0}\quad \approx 8.606\cdot r_{0}}$

Areas in the general case

0) Area of the base semicircle ${\displaystyle A_{0}={\frac {\pi \cdot r_{0}^{2}}{2}}}$
1) Area of the inscribed equilateral triangle ${\displaystyle A_{1}={\frac {1}{\sqrt {3}}}\cdot r_{0}^{2}\quad }$, see Calculation 2

Centroids in the general case

0) By definition the centroid point of a base shape is ${\displaystyle S_{0}=0+0i}$
1) The centroid of the inscribed equilateral triangle relative to the base centroid is: ${\displaystyle S_{1}=\left(0+{\frac {\pi -4}{3\cdot \pi }}\cdot i\right)\cdot r_{0}\quad \approx (0-0.091i)\cdot r_{0}\quad }$, see Calculation 3

Normalised case

In the normalised case the area of the base semicircle is set to 1.
So ${\displaystyle A_{0}={\frac {\pi \cdot r_{0}^{2}}{2}}=1\quad \Rightarrow r_{0}^{2}={\frac {2}{\pi }}\quad \Rightarrow r_{0}={\sqrt {\frac {2}{\pi }}}}$

Segments in the normalised case

0) Radius of the base semicircle ${\displaystyle r_{0}={\sqrt {\frac {2}{\pi }}}\quad \approx 0.798}$
1) Side length of inscribed triangle ${\displaystyle a_{1}={\frac {2}{\sqrt {3}}}\cdot {\sqrt {\frac {2}{\pi }}}={\sqrt {\frac {8}{3\pi }}}\quad \approx 0.921}$

Perimeter in the normalised case

0) Perimeter of base semicircle: ${\displaystyle P_{0}=(2+\pi )\cdot r_{0}=(2+\pi )\cdot {\sqrt {\frac {2}{\pi }}}\quad \approx 4.1023974}$
1) Perimeter of inscribed triangle: ${\displaystyle P_{1}=2{\sqrt {3}}\cdot {\sqrt {\frac {2}{\pi }}}=2{\sqrt {\frac {6}{\pi }}}\quad \approx 2.7639532}$
S) Sum of perimeters: ${\displaystyle P_{s}=P_{0}+P_{1}\quad \approx 6.8663506}$

Areas in the normalised case

0) Area of the base semicircle is by definition ${\displaystyle A_{0}=1}$
1) Area of the inscribed equilateral triangle ${\displaystyle A_{1}={\frac {1}{\sqrt {3}}}\cdot \left({\sqrt {\frac {2}{\pi }}}\right)^{2}={\frac {2}{\pi {\sqrt {3}}}}\quad \approx 0.368}$

Centroids in the normalised case

0) ${\displaystyle S_{0}=0+0i}$
1) ${\displaystyle S_{1}=\left(0+{\frac {\pi -4}{3\cdot \pi }}\cdot i\right)\cdot {\sqrt {\frac {2}{\pi }}}\quad \approx 0-0.073i}$

Distances of centroids

The distance between the centroid of the base semicircle and the centroid of the circle is:
${\displaystyle d_{01}={\sqrt {(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}}}\quad \approx 0.0726712}$
Sum of distances: ${\displaystyle d_{s}=d_{01}\quad \approx 0.0726712}$

Identifying number

Apart of the base element there is only one shape allocated. Therefore the integer part of the identifying number is 1.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.

${\displaystyle decimalpart(6.8663506+0.0726712)=decimalpart(6.9390218)=.9390218}$

So the identifying number is: ${\displaystyle 1.9390218}$

Calculations

Known elements

(0) Given is the radius of the base semicircle ${\displaystyle r_{0}}$
(1) ${\displaystyle \quad |AE|=|CE|=|BE|=r_{0}}$
(2) ${\displaystyle \quad |CD|=|DF|=|CF|=a_{1}}$
(3) ${\displaystyle \quad |DE|=|EF|={\frac {1}{2}}a_{1}}$

Calculation 1

${\displaystyle \quad |CD|^{2}=|DE|^{2}+|CE|^{2}\quad }$, applying the pythagorean theoreme on the rectangular triangle ${\displaystyle \triangle CDE}$
${\displaystyle \Leftrightarrow |CD|^{2}=\left({\frac {1}{2}}a_{1}\right)^{2}+|CE|^{2}\quad }$, applying equation (3)
${\displaystyle \Leftrightarrow |CD|^{2}=\left({\frac {1}{2}}a_{1}\right)^{2}+r_{0}^{2}\quad }$, applying equation (1)
${\displaystyle \Leftrightarrow a_{1}^{2}=\left({\frac {1}{2}}a_{1}\right)^{2}+r_{0}^{2}\quad }$, applying equation (2)
${\displaystyle \Leftrightarrow a_{1}^{2}-\left({\frac {1}{2}}a_{1}\right)^{2}=r_{0}^{2}\quad }$
${\displaystyle \Leftrightarrow a_{1}^{2}-{\frac {1}{4}}a_{1}^{2}=r_{0}^{2}\quad }$
${\displaystyle \Leftrightarrow {\frac {3}{4}}a_{1}^{2}=r_{0}^{2}\quad }$
${\displaystyle \Leftrightarrow a_{1}^{2}={\frac {4}{3}}r_{0}^{2}\quad }$
${\displaystyle \Leftrightarrow a_{1}={\frac {2}{\sqrt {3}}}r_{0}\quad }$

Calculation 2

${\displaystyle \quad A_{1}=|DE|\cdot |CE|}$
${\displaystyle \Leftrightarrow A_{1}={\frac {1}{2}}a_{1}\cdot |CE|\quad }$, applying equation (3)
${\displaystyle \Leftrightarrow A_{1}={\frac {1}{2}}a_{1}\cdot r_{0}\quad }$, applying equation (1)
${\displaystyle \Leftrightarrow A_{1}={\frac {1}{2}}\cdot {\frac {2}{\sqrt {3}}}r_{0}\cdot r_{0}\quad }$, applying Calculation 1
${\displaystyle \Leftrightarrow A_{1}={\frac {1}{\sqrt {3}}}\cdot r_{0}^{2}\quad }$

Calculation 3

${\displaystyle S_{1}=S_{0}+{\overrightarrow {S_{0}E}}+{\overrightarrow {ES_{1}}}}$
${\displaystyle \Leftrightarrow S_{1}=(0+0i)-\left(0+\left({\frac {4\cdot r_{0}}{3\cdot \pi }}\right)\cdot i\right)+(0+{\frac {1}{3}}\cdot |CE|\cdot i)\quad }$, applying the formulas for centroids of triangles and semicircles
${\displaystyle \Leftrightarrow S_{1}=(0+0i)-\left(0+\left({\frac {4\cdot r_{0}}{3\cdot \pi }}\right)\cdot i\right)+(0+{\frac {1}{3}}\cdot r_{0}\cdot i)\quad }$, applying equation (1)
${\displaystyle \Leftrightarrow S_{1}=(0+0+0)+\left(0i-\left({\frac {4\cdot r_{0}}{3\cdot \pi }}\right)\cdot i+{\frac {1}{3}}\cdot r_{0}\cdot i\right)\quad }$, summing the real and the complex terms
${\displaystyle \Leftrightarrow S_{1}=0+\left({\frac {1}{3}}\cdot r_{0}\cdot i-\left({\frac {4\cdot r_{0}}{3\cdot \pi }}\right)\cdot i\right)\quad }$
${\displaystyle \Leftrightarrow S_{1}=0\cdot r_{0}+\left({\frac {1}{3}}\cdot i\cdot r_{0}-{\frac {4}{3\cdot \pi }}\cdot i\cdot r_{0}\right)\quad }$
${\displaystyle \Leftrightarrow S_{1}=0\cdot r_{0}+\left({\frac {1}{3}}-{\frac {4}{3\cdot \pi }}\right)\cdot i\cdot r_{0}\quad }$
${\displaystyle \Leftrightarrow S_{1}=0\cdot r_{0}+\left({\frac {\pi -4}{3\cdot \pi }}\right)\cdot i\cdot r_{0}\quad }$
${\displaystyle \Leftrightarrow S_{1}=\left(0+{\frac {\pi -4}{3\cdot \pi }}\cdot i\right)\cdot r_{0}\quad \approx (0-0.091i)\cdot r_{0}}$

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